Archive | 2019
Cyclotomic and Littlewood Polynomials Associated to Algebras
Abstract
Let A be a finite dimensional algebra over an algebraically closed field k. Assume A is a basic connected and triangular algebra with n pairwise non-isomorphic simple modules. We consider the Coxeter transformation φA T ð Þ as the automorphism of the Grothendieck group K0 A ð Þ induced by the Auslander-Reiten translation τ in the derived category D modA ð Þ of the module category modA of finite dimensional left A-modules. In this paper we study the Mahler measureM χA ð Þ of the Coxeter polynomial χA of certain algebras A. We consider in more detail two cases: (a) A is said to be cyclotomic if all eigenvalues of χA are roots of unity; (b) A is said to be of Littlewood type if all coefficients of χA are 1,0 or 1. We find criteria in order thatA is of one of those types. In particular, we establish new records according to Mossingshoff’s list of Record Mahler measures of polynomials q with 1<M q ð Þ as small as possible, ordered by their number of roots outside the unit circle.